Say I play a Cleric 14/Paladin 6. I have the spell slots of a lvl 17 spellcaster, but can I cast the 9th level spells of the Cleric, or am I limited to 7th? Would this also work with other classes like Sorcerer or Wizard?
The multiclass spellcasting rules are more specific and override the individual class's spellcasting rules.
You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
The example given in that section is a ranger 4/wizard 3. This character knows 3 1st-level ranger spells (as a 4th-level ranger) and has a spellbook with 1st- and 2nd-level wizard spells, of which he can prepare 3 plus his Intelligence modifier (as a 3rd-level wizard).
This example character has 2 3rd-level spell slots, but has no access to 3rd-level spells. He can use those slots to cast 1st- and 2nd-level spells he knows and has prepared, some of which will benefit from the increased level.
Thus, a cleric 14/paladin 6 prepares cleric spells as a 14th-level cleric and paladin spells as a 6th-level paladin.
A 14th-level cleric only has spell slots up to 7th level, meaning you can prepare 14 + Wisdom modifier cleric spells of 7th level and lower.
A 6th-level paladin has spell slots up to 2nd level, so you can prepare 3 + Charisma modifier paladin spells of 2nd level and lower.
Here is a tweet from 5e designer Jeremy Crawford validating this interpretation.
NOTICE: Commentator below has correctly pointed out that the Errata has changed this rule. I'll leave the answer up to prevent similar additional posts in the future.
Josh Clark's answer is correct with one special caveat for Wizards. If you find a level 9 spell scroll or a fellow wizard's spell book and the spell in question has the ritual tag, it's possible. You'd read the spell into your spell book at the cost of 450 gold and 18 hours, while running the risk of messing up and destroying the scroll. The check will be a DC19 vs Arcana, so even with a top level player you'll need to roll well. Consider using inspiration or some other mechanic your DM will allow to gain advantage. This rule can be found on page 114 of the PHB:
"Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it. Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation. For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells."
And the information regarding the DC check can be found on page 200 of the DMG.
A wizard spell on a spell scroll can be copied just as spells in spellbooks can be copied. When a spell is copied from a spell scroll, the copier must succeed on an Intelligence (Arcana) check with a DC equal to 10 + the spell's level. If the check succeeds, the spell is successfully copied. Whether the check succeeds or fails, the spell scroll is destroyed.
Once it's in your spellbook you can still not prepare the spell, however there is technically nothing stopping you from casting the spell as a ritual using your level 9 spell slot. (Page 114 of the PHB):
"Ritual Casting You can cast a wizard spell as a ritual if that spell has the ritual tag and you have the spell in your spellbook. You don't need to have the spell prepared."